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Properties

Label 14.2.26371727281...2032.1
Degree $14$
Signature $[2, 6]$
Discriminant $2^{14}\cdot 3^{12}\cdot 13^{13}$
Root discriminant $55.51$
Ramified primes $2, 3, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $\PGL(2,13)$ (as 14T39)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-108, -648, -1404, 0, 0, 234, 585, 0, 0, 0, -78, 0, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 78*x^10 + 585*x^6 + 234*x^5 - 1404*x^2 - 648*x - 108)
 
gp: K = bnfinit(x^14 - 2*x^13 - 78*x^10 + 585*x^6 + 234*x^5 - 1404*x^2 - 648*x - 108, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} - 78 x^{10} + 585 x^{6} + 234 x^{5} - 1404 x^{2} - 648 x - 108 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2637172728176533939372032=2^{14}\cdot 3^{12}\cdot 13^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.51$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{5}$, $\frac{1}{42} a^{7} + \frac{1}{7} a^{6} - \frac{2}{21} a^{5} - \frac{3}{7} a^{4} - \frac{3}{14} a^{3} - \frac{3}{7} a^{2} - \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{42} a^{8} + \frac{1}{21} a^{6} + \frac{1}{7} a^{5} + \frac{5}{14} a^{4} - \frac{1}{7} a^{3} + \frac{3}{7} a^{2} + \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{84} a^{9} - \frac{1}{14} a^{6} + \frac{23}{84} a^{5} + \frac{5}{14} a^{4} + \frac{3}{7} a^{3} - \frac{1}{2} a^{2} + \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{252} a^{10} + \frac{1}{252} a^{9} - \frac{1}{84} a^{6} - \frac{3}{28} a^{5} - \frac{1}{2} a^{4} + \frac{3}{7} a^{3} - \frac{1}{2} a^{2} - \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{252} a^{11} - \frac{1}{252} a^{9} - \frac{1}{84} a^{7} - \frac{2}{21} a^{6} - \frac{11}{28} a^{5} - \frac{1}{14} a^{4} + \frac{1}{14} a^{3} + \frac{5}{14} a^{2} + \frac{1}{7}$, $\frac{1}{252} a^{12} + \frac{1}{252} a^{9} - \frac{1}{84} a^{8} - \frac{1}{6} a^{6} + \frac{3}{28} a^{5} - \frac{1}{7} a^{4} - \frac{1}{14} a^{3} - \frac{3}{14} a^{2} + \frac{3}{7} a$, $\frac{1}{980028} a^{13} + \frac{323}{490014} a^{12} - \frac{39}{27223} a^{11} + \frac{13}{54446} a^{10} - \frac{13}{108892} a^{9} - \frac{323}{54446} a^{8} - \frac{1783}{163338} a^{7} + \frac{15205}{163338} a^{6} - \frac{15478}{81669} a^{5} - \frac{12959}{54446} a^{4} - \frac{24415}{54446} a^{3} + \frac{7544}{27223} a^{2} - \frac{3907}{27223}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 26696198.5452 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\PGL(2,13)$ (as 14T39):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2184
The 15 conjugacy class representatives for $\PGL(2,13)$
Character table for $\PGL(2,13)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 28 sibling: data not computed
Degree 42 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.13.0.1}{13} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.13.0.1}{13} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.6.6.8$x^{6} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
2.8.8.11$x^{8} + 20 x^{2} + 4$$4$$2$$8$$S_4$$[4/3, 4/3]_{3}^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.13.12.1$x^{13} - 3$$13$$1$$12$$C_{13}:C_3$$[\ ]_{13}^{3}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.13.13.5$x^{13} + 130 x + 13$$13$$1$$13$$F_{13}$$[13/12]_{12}$